On a Balanced System of Linear Equations

概要

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Given a positive integer n, a tri-matrix [numerical formula] is called triply stochastic or 3-balanced if [numerical formula], i = 1, ・・・,n;[numerical formula],j=1,・・・,n;[numerical formula],K=1,・・・,n;u(i,j,k)≥0,i,j, k=1,・・・,n, and if u(i,j1,k1) > 0 and u(i,j2,k2) > 0 imply u(i,j1,k2) > 0 and w(i,j2,k1) > 0. A tri-permutation matrix is a 3-balanced {0,1} tri-matrix A with two permutations π, ρ of the elements of {1,・・・,n} such that u(i, π(I), ρ(I)) = 1 for i = 1,・・・, n. We prove that a matrix is triply stochastic if and only if it is a convex combination of tri-permutation matrices. This result gives a generalization of the well known Birkhoff and von Neumann's theorem on doubly stochastic matrices.
横浜国立大学の論文